Algebra Examples

f (x) = - \sqrt[3]{\frac{2 x + 4}{3}}

$f (x) = - \sqrt[3]{\frac{2 x + 4}{3}}$

Step 1

Write

f (x) = - \sqrt[3]{\frac{2 x + 4}{3}}

$f (x) = - \sqrt[3]{\frac{2 x + 4}{3}}$ as an equation.

y = - \sqrt[3]{\frac{2 x + 4}{3}}

$y = - \sqrt[3]{\frac{2 x + 4}{3}}$

Step 2

Interchange the variables.

x = - \sqrt[3]{\frac{2 y + 4}{3}}

$x = - \sqrt[3]{\frac{2 y + 4}{3}}$

Step 3

Solve for

$y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as

$- \sqrt[3]{\frac{2 y + 4}{3}} = x$ .

$- \sqrt[3]{\frac{2 y + 4}{3}} = x$

Step 3.2

To remove the radical on the left side of the equation, cube both sides of the equation.

${(- \sqrt[3]{\frac{2 y + 4}{3}})}^{3} = x^{3}$

Step 3.3

Simplify each side of the equation.

Tap for more steps...

Step 3.3.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{\frac{2 y + 4}{3}}$ as

${(\frac{2 y + 4}{3})}^{\frac{1}{3}}$ .

${(- {(\frac{2 y + 4}{3})}^{\frac{1}{3}})}^{3} = x^{3}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Simplify

${(- {(\frac{2 y + 4}{3})}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.3.2.1.1

Factor

$2$ out of

$2 y + 4$ .

Tap for more steps...

Step 3.3.2.1.1.1

Factor

$2$ out of

$2 y$ .

${(- {(\frac{2 (y) + 4}{3})}^{\frac{1}{3}})}^{3} = x^{3}$

Step 3.3.2.1.1.2

Factor

$2$ out of

$4$ .

${(- {(\frac{2 y + 2 \cdot 2}{3})}^{\frac{1}{3}})}^{3} = x^{3}$

Step 3.3.2.1.1.3

Factor

$2$ out of

$2 y + 2 \cdot 2$ .

${(- {(\frac{2 (y + 2)}{3})}^{\frac{1}{3}})}^{3} = x^{3}$

Step 3.3.2.1.2

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 3.3.2.1.2.1

Apply the product rule to

$\frac{2 (y + 2)}{3}$ .

${(- \frac{{(2 (y + 2))}^{\frac{1}{3}}}{3^{\frac{1}{3}}})}^{3} = x^{3}$

Step 3.3.2.1.2.2

Apply the product rule to

$2 (y + 2)$ .

${(- \frac{2^{\frac{1}{3}} {(y + 2)}^{\frac{1}{3}}}{3^{\frac{1}{3}}})}^{3} = x^{3}$

Step 3.3.2.1.3

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 3.3.2.1.3.1

Apply the product rule to

$- \frac{2^{\frac{1}{3}} {(y + 2)}^{\frac{1}{3}}}{3^{\frac{1}{3}}}$ .

${(- 1)}^{3} {(\frac{2^{\frac{1}{3}} {(y + 2)}^{\frac{1}{3}}}{3^{\frac{1}{3}}})}^{3} = x^{3}$

Step 3.3.2.1.3.2

Apply the product rule to

$\frac{2^{\frac{1}{3}} {(y + 2)}^{\frac{1}{3}}}{3^{\frac{1}{3}}}$ .

${(- 1)}^{3} \frac{{(2^{\frac{1}{3}} {(y + 2)}^{\frac{1}{3}})}^{3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.3.3

Apply the product rule to

$2^{\frac{1}{3}} {(y + 2)}^{\frac{1}{3}}$ .

${(- 1)}^{3} \frac{{(2^{\frac{1}{3}})}^{3} {({(y + 2)}^{\frac{1}{3}})}^{3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.4

Raise

$- 1$ to the power of

$3$ .

$- \frac{{(2^{\frac{1}{3}})}^{3} {({(y + 2)}^{\frac{1}{3}})}^{3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.5

Simplify the numerator.

Tap for more steps...

Step 3.3.2.1.5.1

Multiply the exponents in

${(2^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.3.2.1.5.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$- \frac{2^{\frac{1}{3} \cdot 3} {({(y + 2)}^{\frac{1}{3}})}^{3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.5.1.2

Cancel the common factor of

$3$ .

Tap for more steps...

Step 3.3.2.1.5.1.2.1

Cancel the common factor.

$- \frac{2^{\frac{1}{3} \cdot 3} {({(y + 2)}^{\frac{1}{3}})}^{3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.5.1.2.2

Rewrite the expression.

$- \frac{2^{1} {({(y + 2)}^{\frac{1}{3}})}^{3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.5.2

Evaluate the exponent.

$- \frac{2 {({(y + 2)}^{\frac{1}{3}})}^{3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.5.3

Multiply the exponents in

${({(y + 2)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.3.2.1.5.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$- \frac{2 {(y + 2)}^{\frac{1}{3} \cdot 3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.5.3.2

Cancel the common factor of

$3$ .

Tap for more steps...

Step 3.3.2.1.5.3.2.1

Cancel the common factor.

$- \frac{2 {(y + 2)}^{\frac{1}{3} \cdot 3}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.5.3.2.2

Rewrite the expression.

$- \frac{2 {(y + 2)}^{1}}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.5.4

Simplify.

$- \frac{2 (y + 2)}{{(3^{\frac{1}{3}})}^{3}} = x^{3}$

Step 3.3.2.1.6

Simplify the denominator.

Tap for more steps...

Step 3.3.2.1.6.1

Multiply the exponents in

${(3^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.3.2.1.6.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$- \frac{2 (y + 2)}{3^{\frac{1}{3} \cdot 3}} = x^{3}$

Step 3.3.2.1.6.1.2

Cancel the common factor of

$3$ .

Tap for more steps...

Step 3.3.2.1.6.1.2.1

Cancel the common factor.

$- \frac{2 (y + 2)}{3^{\frac{1}{3} \cdot 3}} = x^{3}$

Step 3.3.2.1.6.1.2.2

Rewrite the expression.

$- \frac{2 (y + 2)}{3^{1}} = x^{3}$

Step 3.3.2.1.6.2

Evaluate the exponent.

$- \frac{2 (y + 2)}{3} = x^{3}$

Step 3.4

Solve for

$y$ .

Tap for more steps...

Step 3.4.1

Multiply both sides of the equation by

$- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 (y + 2)}{3}) = - \frac{3}{2} x^{3}$

Step 3.4.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.4.2.1

Simplify the left side.

Tap for more steps...

Step 3.4.2.1.1

Simplify

$- \frac{3}{2} (- \frac{2 (y + 2)}{3})$ .

Tap for more steps...

Step 3.4.2.1.1.1

Cancel the common factor of

$3$ .

Tap for more steps...

Step 3.4.2.1.1.1.1

Move the leading negative in

$- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 (y + 2)}{3}) = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.1.2

Move the leading negative in

$- \frac{2 (y + 2)}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 (y + 2)}{3} = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.1.3

Factor

$3$ out of

$- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 (y + 2)}{3} = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 (y + 2)}{3} = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 (y + 2)) = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.2

Cancel the common factor of

$2$ .

Tap for more steps...

Step 3.4.2.1.1.2.1

Factor

$2$ out of

$- 2 (y + 2)$ .

$\frac{- 1}{2} (2 (- (y + 2))) = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- (y + 2))) = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.2.3

Rewrite the expression.

$- - (y + 2) = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.3

Multiply.

Tap for more steps...

Step 3.4.2.1.1.3.1

Multiply

$- 1$ by

$- 1$ .

$1 (y + 2) = - \frac{3}{2} x^{3}$

Step 3.4.2.1.1.3.2

Multiply

$y + 2$ by

$1$ .

$y + 2 = - \frac{3}{2} x^{3}$

Step 3.4.2.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.2.1

Simplify

$- \frac{3}{2} x^{3}$ .

Tap for more steps...

Step 3.4.2.2.1.1

Combine

$x^{3}$ and

$\frac{3}{2}$ .

$y + 2 = - \frac{x^{3} \cdot 3}{2}$

Step 3.4.2.2.1.2

Move

$3$ to the left of

$x^{3}$ .

$y + 2 = - \frac{3 x^{3}}{2}$

Step 3.4.3

Subtract

$2$ from both sides of the equation.

$y = - \frac{3 x^{3}}{2} - 2$

Step 4

Replace

$y$ with

$f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = - \frac{3 x^{3}}{2} - 2$

Step 5

Verify if

$f^{- 1} (x) = - \frac{3 x^{3}}{2} - 2$ is the inverse of

$f (x) = - \sqrt[3]{\frac{2 x + 4}{3}}$ .

Tap for more steps...

Step 5.1

To verify the inverse, check if

$f^{- 1} (f (x)) = x$ and

$f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate

$f^{- 1} (f (x))$ .

Tap for more steps...

Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}})$ by substituting in the value of

$f$ into

$f^{- 1}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 {(- \sqrt[3]{\frac{2 x + 4}{3}})}^{3}}{2} - 2$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Factor

$2$ out of

$2 x + 4$ .

Tap for more steps...

Step 5.2.3.1.1

Factor

$2$ out of

$2 x$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 {(- \sqrt[3]{\frac{2 (x) + 4}{3}})}^{3}}{2} - 2$

Step 5.2.3.1.2

Factor

$2$ out of

$4$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 {(- \sqrt[3]{\frac{2 x + 2 \cdot 2}{3}})}^{3}}{2} - 2$

Step 5.2.3.1.3

Factor

$2$ out of

$2 x + 2 \cdot 2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 {(- \sqrt[3]{\frac{2 (x + 2)}{3}})}^{3}}{2} - 2$

Step 5.2.3.2

Simplify the numerator.

Tap for more steps...

Step 5.2.3.2.1

Apply the product rule to

$- \sqrt[3]{\frac{2 (x + 2)}{3}}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 {(- 1)}^{3} {\sqrt[3]{\frac{2 (x + 2)}{3}}}^{3}}{2} - 2$

Step 5.2.3.2.2

Raise

$- 1$ to the power of

$3$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {\sqrt[3]{\frac{2 (x + 2)}{3}}}^{3})}{2} - 2$

Step 5.2.3.2.3

Rewrite

$\sqrt[3]{\frac{2 (x + 2)}{3}}$ as

$\frac{\sqrt[3]{2 (x + 2)}}{\sqrt[3]{3}}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)}}{\sqrt[3]{3}})}^{3})}{2} - 2$

Step 5.2.3.2.4

Multiply

$\frac{\sqrt[3]{2 (x + 2)}}{\sqrt[3]{3}}$ by

$\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)}}{\sqrt[3]{3}} \cdot \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}})}^{3})}{2} - 2$

Step 5.2.3.2.5

Combine and simplify the denominator.

Tap for more steps...

Step 5.2.3.2.5.1

Multiply

$\frac{\sqrt[3]{2 (x + 2)}}{\sqrt[3]{3}}$ by

$\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{\sqrt[3]{3} {\sqrt[3]{3}}^{2}})}^{3})}{2} - 2$

Step 5.2.3.2.5.2

Raise

$\sqrt[3]{3}$ to the power of

$1$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{\sqrt[3]{3} {\sqrt[3]{3}}^{2}})}^{3})}{2} - 2$

Step 5.2.3.2.5.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1 + 2}})}^{3})}{2} - 2$

Step 5.2.3.2.5.4

Add

$1$ and

$2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{3}})}^{3})}{2} - 2$

Step 5.2.3.2.5.5

Rewrite

${\sqrt[3]{3}}^{3}$ as

$3$ .

Tap for more steps...

Step 5.2.3.2.5.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{3}$ as

$3^{\frac{1}{3}}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{{(3^{\frac{1}{3}})}^{3}})}^{3})}{2} - 2$

Step 5.2.3.2.5.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{3^{\frac{1}{3} \cdot 3}})}^{3})}{2} - 2$

Step 5.2.3.2.5.5.3

Combine

$\frac{1}{3}$ and

$3$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}})}^{3})}{2} - 2$

Step 5.2.3.2.5.5.4

Cancel the common factor of

$3$ .

Tap for more steps...

Step 5.2.3.2.5.5.4.1

Cancel the common factor.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}})}^{3})}{2} - 2$

Step 5.2.3.2.5.5.4.2

Rewrite the expression.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{3})}^{3})}{2} - 2$

Step 5.2.3.2.5.5.5

Evaluate the exponent.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} {\sqrt[3]{3}}^{2}}{3})}^{3})}{2} - 2$

Step 5.2.3.2.6

Simplify the numerator.

Tap for more steps...

Step 5.2.3.2.6.1

Rewrite

${\sqrt[3]{3}}^{2}$ as

$\sqrt[3]{3^{2}}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} \sqrt[3]{3^{2}}}{3})}^{3})}{2} - 2$

Step 5.2.3.2.6.2

Raise

$3$ to the power of

$2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2)} \sqrt[3]{9}}{3})}^{3})}{2} - 2$

Step 5.2.3.2.7

Simplify the numerator.

Tap for more steps...

Step 5.2.3.2.7.1

Combine using the product rule for radicals.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{2 (x + 2) \cdot 9}}{3})}^{3})}{2} - 2$

Step 5.2.3.2.7.2

Multiply

$9$ by

$2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 {(\frac{\sqrt[3]{18 (x + 2)}}{3})}^{3})}{2} - 2$

Step 5.2.3.2.8

Apply the product rule to

$\frac{\sqrt[3]{18 (x + 2)}}{3}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{{\sqrt[3]{18 (x + 2)}}^{3}}{3^{3}})}{2} - 2$

Step 5.2.3.2.9

Simplify the numerator.

Tap for more steps...

Step 5.2.3.2.9.1

Rewrite

${\sqrt[3]{18 (x + 2)}}^{3}$ as

$18 (x + 2)$ .

Tap for more steps...

Step 5.2.3.2.9.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{18 (x + 2)}$ as

${(18 (x + 2))}^{\frac{1}{3}}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{{({(18 (x + 2))}^{\frac{1}{3}})}^{3}}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{{(18 (x + 2))}^{\frac{1}{3} \cdot 3}}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.1.3

Combine

$\frac{1}{3}$ and

$3$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{{(18 (x + 2))}^{\frac{3}{3}}}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.1.4

Cancel the common factor of

$3$ .

Tap for more steps...

Step 5.2.3.2.9.1.4.1

Cancel the common factor.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{{(18 (x + 2))}^{\frac{3}{3}}}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.1.4.2

Rewrite the expression.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{18 (x + 2)}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.1.5

Simplify.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{18 (x + 2)}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.2

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{18 x + 18 \cdot 2}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.3

Multiply

$18$ by

$2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{18 x + 36}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.4

Factor

$18$ out of

$18 x + 36$ .

Tap for more steps...

Step 5.2.3.2.9.4.1

Factor

$18$ out of

$18 x$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{18 (x) + 36}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.4.2

Factor

$18$ out of

$36$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{18 x + 18 \cdot 2}{3^{3}})}{2} - 2$

Step 5.2.3.2.9.4.3

Factor

$18$ out of

$18 x + 18 \cdot 2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{18 (x + 2)}{3^{3}})}{2} - 2$

Step 5.2.3.2.10

Raise

$3$ to the power of

$3$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{18 (x + 2)}{27})}{2} - 2$

Step 5.2.3.2.11

Cancel the common factor of

$18$ and

$27$ .

Tap for more steps...

Step 5.2.3.2.11.1

Factor

$9$ out of

$18 (x + 2)$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{9 (2 (x + 2))}{27})}{2} - 2$

Step 5.2.3.2.11.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.2.11.2.1

Factor

$9$ out of

$27$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{9 (2 (x + 2))}{9 (3)})}{2} - 2$

Step 5.2.3.2.11.2.2

Cancel the common factor.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{9 (2 (x + 2))}{9 \cdot 3})}{2} - 2$

Step 5.2.3.2.11.2.3

Rewrite the expression.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{3 \cdot (- 1 \frac{2 (x + 2)}{3})}{2} - 2$

Step 5.2.3.2.12

Combine exponents.

Tap for more steps...

Step 5.2.3.2.12.1

Factor out negative.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- (3 (\frac{2 (x + 2)}{3}))}{2} - 2$

Step 5.2.3.2.12.2

Combine

$3$ and

$\frac{2 (x + 2)}{3}$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- \frac{3 (2 (x + 2))}{3}}{2} - 2$

Step 5.2.3.2.12.3

Multiply

$2$ by

$3$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- \frac{6 (x + 2)}{3}}{2} - 2$

Step 5.2.3.2.13

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.3.2.13.1

Reduce the expression

$\frac{6 (x + 2)}{3}$ by cancelling the common factors.

Tap for more steps...

Step 5.2.3.2.13.1.1

Factor

$3$ out of

$6 (x + 2)$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- \frac{3 (2 (x + 2))}{3}}{2} - 2$

Step 5.2.3.2.13.1.2

Factor

$3$ out of

$3$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- \frac{3 (2 (x + 2))}{3 (1)}}{2} - 2$

Step 5.2.3.2.13.1.3

Cancel the common factor.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- \frac{3 (2 (x + 2))}{3 \cdot 1}}{2} - 2$

Step 5.2.3.2.13.1.4

Rewrite the expression.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- \frac{2 (x + 2)}{1}}{2} - 2$

Step 5.2.3.2.13.2

Divide

$2 (x + 2)$ by

$1$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- (2 (x + 2))}{2} - 2$

Step 5.2.3.3

Multiply

$- 1$ by

$2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- 2 (x + 2)}{2} - 2$

Step 5.2.3.4

Cancel the common factor of

$- 2$ and

$2$ .

Tap for more steps...

Step 5.2.3.4.1

Factor

$2$ out of

$- 2 (x + 2)$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{2 (- (x + 2))}{2} - 2$

Step 5.2.3.4.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.4.2.1

Factor

$2$ out of

$2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{2 (- (x + 2))}{2 (1)} - 2$

Step 5.2.3.4.2.2

Cancel the common factor.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{2 (- (x + 2))}{2 \cdot 1} - 2$

Step 5.2.3.4.2.3

Rewrite the expression.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - \frac{- (x + 2)}{1} - 2$

Step 5.2.3.4.2.4

Divide

$- (x + 2)$ by

$1$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = (x + 2) - 2$

Step 5.2.3.5

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - (- x - 1 \cdot 2) - 2$

Step 5.2.3.6

Multiply

$- 1$ by

$2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = - (- x - 2) - 2$

Step 5.2.3.7

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = x + 2 - 2$

Step 5.2.3.8

Multiply

$- - x$ .

Tap for more steps...

Step 5.2.3.8.1

Multiply

$- 1$ by

$- 1$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = 1 x + 2 - 2$

Step 5.2.3.8.2

Multiply

$x$ by

$1$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = x + 2 - 2$

Step 5.2.3.9

Multiply

$- 1$ by

$- 2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = x + 2 - 2$

Step 5.2.4

Combine the opposite terms in

$x + 2 - 2$ .

Tap for more steps...

Step 5.2.4.1

Subtract

$2$ from

$2$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = x + 0$

Step 5.2.4.2

Add

$x$ and

$0$ .

$f^{- 1} (- \sqrt[3]{\frac{2 x + 4}{3}}) = x$

Step 5.3

Evaluate

$f (f^{- 1} (x))$ .

Tap for more steps...

Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate

$f (- \frac{3 x^{3}}{2} - 2)$ by substituting in the value of

$f^{- 1}$ into

$f$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{2 (- \frac{3 x^{3}}{2} - 2) + 4}{3}}$

Step 5.3.3

Factor

$2$ out of

$2 (- \frac{3 x^{3}}{2} - 2) + 4$ .

Tap for more steps...

Step 5.3.3.1

Factor

$2$ out of

$4$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{2 (- \frac{3 x^{3}}{2} - 2) + 2 \cdot 2}{3}}$

Step 5.3.3.2

Factor

$2$ out of

$2 (- \frac{3 x^{3}}{2} - 2) + 2 \cdot 2$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{2 (- \frac{3 x^{3}}{2} - 2 + 2)}{3}}$

Step 5.3.4

Add

$- 2$ and

$2$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{2 (- \frac{3 x^{3}}{2} + 0)}{3}}$

Step 5.3.5

Add

$- \frac{3 x^{3}}{2}$ and

$0$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{2 \cdot (- 1 \frac{3 x^{3}}{2})}{3}}$

Step 5.3.6

Combine exponents.

Tap for more steps...

Step 5.3.6.1

Factor out negative.

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- (2 (\frac{3 x^{3}}{2}))}{3}}$

Step 5.3.6.2

Combine

$2$ and

$\frac{3 x^{3}}{2}$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- \frac{2 (3 x^{3})}{2}}{3}}$

Step 5.3.6.3

Multiply

$3$ by

$2$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- \frac{6 x^{3}}{2}}{3}}$

Step 5.3.7

Cancel the common factor of

$6$ and

$2$ .

Tap for more steps...

Step 5.3.7.1

Factor

$2$ out of

$6 x^{3}$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- \frac{2 (3 x^{3})}{2}}{3}}$

Step 5.3.7.2

Cancel the common factors.

Tap for more steps...

Step 5.3.7.2.1

Factor

$2$ out of

$2$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- \frac{2 (3 x^{3})}{2 (1)}}{3}}$

Step 5.3.7.2.2

Cancel the common factor.

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- \frac{2 (3 x^{3})}{2 \cdot 1}}{3}}$

Step 5.3.7.2.3

Rewrite the expression.

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- \frac{3 x^{3}}{1}}{3}}$

Step 5.3.7.2.4

Divide

$3 x^{3}$ by

$1$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- (3 x^{3})}{3}}$

Step 5.3.8

Cancel the common factor of

$3$ .

Tap for more steps...

Step 5.3.8.1

Cancel the common factor.

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{\frac{- (3 x^{3})}{3}}$

Step 5.3.8.2

Divide

$- (x^{3})$ by

$1$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{- (x^{3})}$

Step 5.3.9

Rewrite

$- x^{3}$ as

${(- x)}^{3}$ .

$f (- \frac{3 x^{3}}{2} - 2) = - \sqrt[3]{{(- x)}^{3}}$

Step 5.3.10

Pull terms out from under the radical, assuming real numbers.

$f (- \frac{3 x^{3}}{2} - 2) = x$

Step 5.4

Since

$f^{- 1} (f (x)) = x$ and

$f (f^{- 1} (x)) = x$ , then

$f^{- 1} (x) = - \frac{3 x^{3}}{2} - 2$ is the inverse of

$f (x) = - \sqrt[3]{\frac{2 x + 4}{3}}$ .

$f^{- 1} (x) = - \frac{3 x^{3}}{2} - 2$